Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits

Indranil Biswas; Junho Peter Whang; Mahan Mj; Subhojoy Gupta

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Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits

Indranil Biswas, Subhojoy Gupta, Mahan Mj, Junho Peter Whang

Geometric Topology · Submitted 1 Jul 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface.

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arXiv ID1707.00071 (version 4)

Pith IDGX7U2EYW, Pith's permanent record of this paper and its review

CitationIndranil Biswas, Junho Peter Whang, Mahan Mj, Subhojoy Gupta. "Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits." arXiv:1707.00071 (2017). https://pith.science/paper/GX7U2EYW

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,551 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface.	contribution 0.93	0.94	0.31	asserted 0.99
s1	For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image.	result 0.98	0.79	0.24	asserted 1.00
s2	For genus one, they correspond to the finite or special dihedral representations.	result 1.00	0.80	0.12	asserted 1.00
s3	We also obtain an analogous result for bounded orbits in the moduli space.	result 0.83	0.93	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v4

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.

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4 arXiv papers cite this in 9 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Cites it as support 1 paper

No such representations exist by [ref] .

Canonical representations of surface groups · Bounds and examples · 0.98

Cites it as background 1 paper

The beautiful paper [ref] handles the $2$-dimensional case ($r = 2, g ≥1, n ≥0$) with trivial determinant.

Canonical representations of surface groups · MCG-finite representations and Painlevé VI · 0.81

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